A With-In Host Dengue Infection Model with Immune Response and Beddington-DeAngelis Incidence Rate ()
1. Introduction
Dengue is an infections mosquito-borne viral disease. It is estimated that about 50 million infections occur annually in over 100 countries [1]. There is no specific treatment for curingdengue patients. Hospital treatment in general is given as supportive care which includes bed rest, antipyretics, and analgesics. Most dengue infections are asymptomatic. Few of them suffer dengue fever and dengue haemorrhagic fever, which may end up in fatality.
Dengue virus is one of the most difficult arboviruses to isolate. There are four serotypes of the dengue virus and each of the serotype has numerous virus strains. Infection with one dengue serotype may provide lifelong immunity to that serotype, but there is no cross-protective immunity to other serotype, [2]. Identification of the primary target cells of dengue virus replication in infected human body has proven to be extremely difficult. It is generally believed that the target cells of dengue virus are monocytes or its differentiated cells the macrophages [3].
It is usually believed that dengue virus is quickly cleared in human body within approximately 7 days after the day of sudden onset of fever [1]. Naturally this clearing process is done by the immune system which is a result of complex dynamic reactions. Following [4], in this paper we try to understand the process using a mathematical model.
Mathematical modeling of dengue disease transmission in human and mosquito populations has been done since the beginning of last century. Some of the recent models could be seen in [2-5]. Several studies on infection model within human body have been done for various cases [2,3] and [5-11]. Meanwhile, mathematical modeling for with-in host dengue viral disease is quite new.
The model for with-in host dengue viral infection with Beddington-DeAngelis incidence rate and immune response is as following.
(1)
where, 
and
.
The constant
, is the rate constant characterizing infection of the cells. The constants 
are positive.
In the above 
and 
represent the density of susceptible monocytes, infected monocytes, free virus particles and immune cells in 1 μl blood at time t, respective. The production of susceptible monocytes by bone marrow is assumed at a constant rate μ and the life span of susceptible monocytes is
. The flow from susceptible monocytes to the infected monocytes depends on the incidence rate of susceptible monocytes and free virus particle. This rate is shown by 
where 
is the incidence response of susceptible monocytes to free virus particles. The period of infected monocytes is assumed constant as 
. We assumed virus multiplication is at constant rate k and the virus clearance rate is at constant rate
. We also assumed the immune cells are produced at constant rate 
and their life span is
. Moreover we assumed there is stimulation of immune cells production due to the increase of infected cell which is proportional to the density infected monocytes at a constant rate c as well as from the contacts with infected cells at the rate d and the immune cells will eliminate the infected monocytes at a constant rate v. Finally, the positive constants 
and 
have some biological meanings.
The above model is valid for only one serotype of dengue virus circulate in an infected host and dengue infects monocytes in blood stream.
For more detail the reader is referred to [4] and references therein.
The local stability of the equilibrium points of the system (1) for Lotka-Voltera functional response i.e. 
, has been discussed in [4]. The model (1) is a generalization of the self-regulating cytotoxic T lymphocytes (CTL) response model. The predator-prey like CTL response model and the linear immune response model in chapter 6 of [5].
In this paper, we will analyze the global of stability of the viral free equilibrium for Beddington-DeAngelis incidence response,
. In fact we will show that if this equilibrium is the only rest point of the system (1), then it is globally asymptotically stable. If there are some other equilibria, then the local stability of them depends on the values of the parameters.
2. Global Stability of the Uninfected Equilibrium
In this section, at first we will find the equilibrium points of the system (1) and the eigenvalues of this system at these points. This information leads us to prove the locally asymptotical stability of the equilibrium points.
At an equilibrium point of the system (1) we must have
(2)
From the first equation we obtain,
.
Substituting this value of V into the third equation yields, 
. From the fourth equation we obtain 
. Substituting these values of 
and 
into the second equation yields,
If, 
, then from this, we have
. Thus 
is one of the equilibrium points of the system (1). If,
then
(3)
where,
,
and
In the following we consider the stability property of the equilibrium point
. In order to do this we check the sign of the eigenvalues of Jacobi matrix of (1) at
. The Jacobi matrix is
(4)
So the value of 
at 
is
The eigenvalues of 
are the roots of the characteristic polynomial
Thus, 
and 
are two of the eigenvalues and the other two are the roots of
.
These roots are
where,
.
Clearly, 
, 
and
have negative real part. If 
has negative real part, then the equilibrium 
is locally asymptotically stable. But 
is negative if and only if,
. This condition equals to,
.
Set,
. This number is called the basic reproduction ratio [7].
Therefore we have the following theorem.
Theorem 2.1. If, 
, the equilibrium point 
is locally asymptotically stable and if
, the equilibrium 
is unstable.
Now we will show that if, 
, then the equilibrium, 
is globally asymptotically stable. In order to see this, first of all consider the following domain in the 
space.
It follows that the flow generated by that system (1.1) gets into 
on the boundary of
. Let 
for,
. Thus 
is a global attractor. Now in 
consider the following set for
:
where, 
and
and
.
If we differentiate 
along the orbits of the system (1), we obtain:
Here,
. Since on the surface 
of the boundary of
, we have 
and 
and 
, therefore
. Thus the flow gets into 
on,
. Hence the flow gets into 
from its boundary. Therefore 
is an attractor in D for all
. But
. Thus 
is a global attractor. Thus we have proved the following theorem.
Theorem 2.2. If, 
, then 
,the uninfected equilibrium is the only equilibrium of the system (1). Moreover this equilibrium is globally asymptotically stable.
Since 
is globally asymptotically stable for
, any other equilibrium points of the system (1) cannot exist for
. Therefore, 
is the unique equilibrium point for
.
3. Stability of the Other Equilibrium Points
In this section, we consider the stability of the other rest point of the system (1). In order to this, we consider the Equation (3). First, we consider this equation for 
and then for
.
There are two cases for 
as follows.
Case 1. 
In this case, the system (1) has two equilibrium points, 
and another one. To see this, from the first equation of (2) we obtain,
. Since 
from the fourth equation we get
. Substituting these values of 
and 
into the second equation yields,
. By using the value of 
and the third equation we get
. Using these values into the equation (3), we obtain,
.
Therefore, we obtain
where
as the second rest point of the system (1).
Notice that this rest point exists if
, or
. If
, this rest point is the same as
. If
, then
or
.
Now, we consider the local stability of the equilibrium
. By using the formula (4), the value of Jacobi matrix at 
is
where
, 
,
and
.
We calculate the eigenvalues of 
as follow:
Thus one of the roots is
. The other roots are given by
Here
, 
and 
.
By substituting the value of 
and 
in 
and 
we see that 
and 
are positive. Moreover, it is easy to check that,
. By the Rouths Hurwitz Criteria, all roots of the cubic polynomial have negative real part. Therefore we have the following theorem.
Theorem 3.1. If, 
, then the equilibrium point 
exists and is locally asymptotically stable. Moreover the equilibrium 
exists and is unstable.
Remark 3.1. Since, the rest points and the eigenvalues depend continuously on the parameters, thus for small values of 
and
, 
exists and is locally asymptotically stable.
Case 2. 
In this case, the system (1) has three equilibrium points and 
is one of them. Since
, the fourth equation of the system (1) gives 
Therefore, 
or 
For
, substituting the value of 
into the second equation yields,
. By using the value of 
and the third equation we get,
. Substituting values of 
and 
into the equation (3), we obtain,
. Thus, we get
as another rest point of the system (1).
For
, by substituting this value of 
in the second equation of the system (1), we obtain
. Then from the third equation we get,
. By using this value of 
into the first equation of the system (1), we obtain the following quadratic equation.
If 
and
then 
is another equilibrium point of the system (1) where 
is the positive root of the above quadratic equation.
In the following, we consider the stability property of these points.
At first consider it for
. Here we check the sign of the eigenvalues of Jacobi matrix of the system (1) at
. From the formula (4) we have
where
, 
,
and
.
We calculate the eigenvalues of 
as follows:
Thus one of the roots is
. The other roots are given by
Here
, 
and 
.
By substituting the value of 
and 
in 
and 
we will see that 
and 
are positive. Moreover, it is easy to see that,
. By the Rouths Hurwitz Criteria, all roots of the cubic polynomial have negative real part. If 
or
, then real part of all of the eigenvalues are negative. Therefore the point 
is locally asymptotically stable.
Now we consider the stability property of the other equilibrium point,
. From the formula (4) we have
where
, 
, 
, 
and 
.
The eigenvalues of the matrix 
are given by the algebraic equation,
or
Then from the above equation we get
By considering the value of 
it follows that all of the coefficients of the above equation are positive, then from Routh Hourwitz Criteria we see that all of the roots have negative real parts. Therefore we have the following theorem.
Theorem 3.2. For 
and
, we have the following results.
1) If
, the equilibrium points 
and 
are the only rest points of the system (1), then 
is unstable and 
is locally asymptotically stable.
2) If
, then for
, the equilibrium
, is locally asymptotically stable and for
, it becomes unstable. If
,
and 
are the only two rest points of the system (1) and 
does not exist.
3) If 
and
then the equilibrium 
exists and is locally asymptotically stable. Moreover the equilibrium points 
and 
are unstable.
Remark 3.2. If
, the point 
does not exist, therefore the point 
is the only endemic equilibrium point of the system (1). Also, for 
and
, the point 
is the only endemic equilibrium point.
Remark 3.3. 1) From continuous dependent of the equilibrium points and eigenvalues to the parameters, Theorem 3.1 and 3.2 must be valued for 
and small.
2) For the case, 
and large, if
, the point
is the unique equilibrium of the system
(1) which is globally asymptotically stable. If
, the system (1) has a unique endemic equilibrium point, 
satisfying in the equations
,
, 
and the equation (3). Here stability property of this point is not shown.
4. Numerical Simulation
For the following numerical simulations, we use parameters of T-cells as the parameters of immune cells, those are
, 
days. The estimated value of 
is obtained by assuming that the equilibrium value of the density of immune cells in the absence of infection is 2000 cells.
In this model the endemic status of the disease depends on the individual response toward incoming viruses. The larger the invasion rate
, the chance is higher to catch the disease. On the contrary the increase of the elimination rate 
of infected cell, the risk of infection is lower.
For
,
,
, we have For 
we obtain the same result in the above table.
If 
then for the same value of parameters we have the following table.
5. Conclusions
In order to understand the main characteristic of Dengue mystery, the author in [4] assumed that this virus can be eliminated by immune response which is described by the last equation of the system (1).
By using linear incidence rate of susceptible and free virus particle, they analyzed the existence of the endemic virus equilibria.
In this paper, from the analysis of the endemic equilibria it is found that, for Beddington DeAngelis incidence rate of susceptible and free virus particle, the same results are valid.
The reson for this correspondence is that in both models, the feature of the immune response is described by the term
. However, the parameter 
in Beddington DeAngles makes the elimination of dengue virus by immune response in a shorter time. This fact can be seen by comparing Tables 1 and 2.
Table 1. status of equilibrium points of system (1) in the case
.
Table 2. status of equilibrium points of system (1) in the case
,
.
By Theorem 2.2, if the basic reproduction number, 
is less than one, then uninfected equilibrium point, 
is the only steady state point of system (1) and it is globally asymptotically state. This means that the virus is eliminated by immune response. For larger values of 
and
, 
is more attractor and the virus is cleared much faster.
If the basic reproduction number, 
is more than one; for
, besides of the uninfected steady state 
which is uninfected, there are some infected steady state. Here we consider two cases of endemic virus.
Fist, for
, we have only one infected endemic
. If
, there is no immune response, so the density of susceptible moncytes equal zero. In
, this density equals
, so it does not depend on the other parameters for virus load of infected cell. For lager values of 
and
, the infected endemic 
is closer to the uninfected endemic 
and it is more controllable.
Second, for 
and
, from Theorem 3.2 we see that if 
is negative or positive small, then there is only one infected endemic equilibrium 
which is stable. However if 
is positive and large, then the endemic virus equilibrium 
exists and is stable. This means that we found 
new threshold for
. For condition 
is less than this threshold the dynamic of the model is qualitatively same as the case
. When 
is greater than this threshold, we have a new endemic virus equilibrium, 
which is stable and the equilibrium points 
and 
are unstable. From the components of the endemic equilibrium 
we see that after the onset of the symptom, if 
increases, the 
and 
components of equilibria decrease and the 
and Z-components of equilibria will increase. Conversely, if 
and 
increase, the 
and I-components of equilibria will decrease but the virus load increases at the initial viral infection.
For case 
and large and
, the model has a unique endemic virus. The 
and 
components of this equilibrium point decrease as 
increases and the 
and Z-components of it increase as 
increases.
Therefore, 
and 
are the important parameters to capture the phenomena that dengue virus is quickly cleared in 
shorter time.
6. Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.